Optimal Transport Networks in Spatial Equilibrium Pablo D. - - PowerPoint PPT Presentation

Optimal Transport Networks in Spatial Equilibrium

Pablo D. Fajgelbaum Edouard Schaal

UCLA/NBER, CREI/CEPR

World Bank Space and Productivity Conference, 09/2017

Introduction

How should transport infrastructure investments be allocated in a network? Link-by-link

◮ Direct effect: value of trade/travel time along a link ◮ Indirect effect: reallocations, investment,..

Trade-offs across links

◮ Shape of network will matter

This Paper

Develop a framework to study optimal transport networks in general equilibrium Neoclassical trade model (with factor mobility) on a graph Sub-Problems

◮ how to allocate production and consumption? ◮ how to ship goods through the network? (“Optimal Flows”) ◮ how to build infrastructure? (“Optimal Network”)

Apply to road networks in different European countries

◮ how large are losses from misallocation of current networks and gains from optimal

expansion?

◮ how do these effects vary across countries? ◮ what are the regional effects?

Brief Background

Trade costs in quantitative analyses of international trade and economic geography

◮ Eaton and Kortum (2002), Allen and Arkolakis (2014), Redding (2016),...

Assessments of actual changes in transport infrastructure

◮ Donaldson (2010), Duranton et al. (2014), Faber (2014),...

Standard approach is to assess implications of a particular (not necessarily “best”) change in transport costs Here: trade costs are endogenous through optimal investments in the transport network

◮ Related to optimal flow problem on a network (e.g., Chapter 8 of Galichon, 2016)

Framework

1

Multiple locations and factors of production

◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2

The locations are arranged on a graph

◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Qn jk can flow through neighbors only 3

Per-unit cost τ n

jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure Ijk 4

Building infrastructure Ijk takes up δI

jkIjk units of a scarce resource (“asphalt”) ◮ δI jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by {Ijk}

Framework

1

Multiple locations and factors of production

◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2

The locations are arranged on a graph

◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Qn jk can flow through neighbors only 3

Per-unit cost τ n

jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure Ijk 4

Building infrastructure Ijk takes up δI

jkIjk units of a scarce resource (“asphalt”) ◮ δI jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by {Ijk}

Framework

1

Multiple locations and factors of production

◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2

The locations are arranged on a graph

◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Qn jk can flow through neighbors only 3

Per-unit cost τ n

jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure Ijk 4

Building infrastructure Ijk takes up δI

jkIjk units of a scarce resource (“asphalt”) ◮ δI jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by {Ijk}

Framework

1

Multiple locations and factors of production

◮ Many traded goods and one non-traded good ◮ Neoclassical production technologies ◮ Workers may be immobile or perfectly immobile ⋆ Homothetic preferences of traded and non-traded commodities ◮ Special cases: Ricardian model, Heckscher-Ohlin, Armington, Rosen-Roback... 2

The locations are arranged on a graph

◮ Each location has a set of “neighbors” (directly connected) ⋆ “Neighbors” may be geographically distant ◮ Shipments Qn jk can flow through neighbors only 3

Per-unit cost τ n

jk of shipping from j to a neighbor k ◮ Decreasing returns to transport: τ n jk increases with quantity shipped ◮ Positive returns to infrastructure: τ n jk decreases with infrastructure Ijk 4

Building infrastructure Ijk takes up δI

jkIjk units of a scarce resource (“asphalt”) ◮ δI jk may vary across links, due to ruggedness, distance... ◮ The transport network is defined by {Ijk}

Example: Spain

50 km x 50 km square network, 8 neighbors per interior node The problem of designing the network determines how much to invest in each link

Optimal Network Problem

Assume that, given the transport network, the market efficiently allocates resources

◮ Requires tolls if the decreasing returns to transport are not internalized ◮ Free entry to trading activities

Planning problem: choose {Ijk} to maximize aggregate welfare subject to:

detail ◮ Location and consumption decisions by workers given

• Ijk
• ◮ Production and trade decisions of firms given
• Ijk
• ◮ Goods and factors market clearing constraints

◮ Availability of inputs to build the network

Key Efficiency Conditions on Every Link

No-arbitrage condition on flows (“efficient road use”): Pn

k

Pn

j

≤ 1 + τ n

jk +

∂τ n

jk

∂Qn

jk

Qn

jk, = if Qn jk > 0

Optimal investment:

µδI

jk

• Building Cost

≥

• n

Pn

j Qn jk

• −

∂τ n

jk

∂Ijk

• Gain from Infrastructure

, = if Ijk > 0

Full problem is globally convex under strong enough congestion in transport

Key Efficiency Conditions on Every Link

No-arbitrage condition on flows (“efficient road use”): Pn

k

Pn

j

≤ 1 + τ n

jk +

∂τ n

jk

∂Qn

jk

Qn

jk, = if Qn jk > 0

Optimal investment:

µδI

jk

• Building Cost

≥

• n

Pn

j Qn jk

• −

∂τ n

jk

∂Ijk

• Gain from Infrastructure

, = if Ijk > 0

Full problem is globally convex under strong enough congestion in transport

Key Efficiency Conditions on Every Link

No-arbitrage condition on flows (“efficient road use”): Pn

k

Pn

j

≤ 1 + τ n

jk +

∂τ n

jk

∂Qn

jk

Qn

jk, = if Qn jk > 0

Optimal investment:

µδI

jk

• Building Cost

≥

• n

Pn

j Qn jk

• −

∂τ n

jk

∂Ijk

• Gain from Infrastructure

, = if Ijk > 0

Full problem is globally convex under strong enough congestion in transport

Example: Uniform Geography

20 randomly placed cities across uniform geography

Building Cost: δI

jk = δ0Distanceδ1 jk

Example: Role of Geography

Convex case

Building Cost: δI

jk = δ0Distanceδ1 jk

• 1 + |∆Elevation|jk

δ2 δ

CrossingRiverjk 3

δ

AlongRiverjk 4

nonconvex

Application to European Countries

In 25 European countries we observe:

◮ Road networks (EuroGeographics) ◮ Value Added (G-Econ 4.0) ◮ Population (GPW)

Parametrization

◮ Each locations in the model = population centroid of 0.5 x 0.5 degree (∼50 km) cell ◮ Observed infrastructure I obs

jk

assigned to reproduce actual network

map ◮ Each location produces one traded and one non-traded good ⋆ Productivity in traded sector and endowment of non-traded good calibrated to

match value added and population

◮ Transport technology: τ n

jk = δτ jk

• Qn

jk

β I γ

jk

, benchmark γ = β

◮ Building costs δI

jk as function of distance and ruggedness from Collier et al. (2016),

based on WB’s ROCKS.

Losses from Misallocation

Simulate counterfactual optimal network for each country

◮ How much real income is lost in each country due to misplacement of existing roads? AT BE CY CZ DK FI FR GE DE HU IE IT LV LT LU MK MD NL ND PT RS SK SI ES CHLI AT BE CY CZ DK FI FR GE DE HU IE IT LV LT LU MK MD NL ND PT RS SK SI ES CHLI

5 10 15 20 % Gain 7 8 9 10 11 Log Income Per Capita Fixed Labor Mobile Labor

Linear regression slope (robust SE): Mobile Labor: -2.523 (1.269); Fixed Labor: -2.035 (.712)

Average: 4.6% (fixed labor); 4.8% (mobile labor).

Where should infrastructure be placed?

“Back of the envelope” for how optimal investments should be undertaken

◮ Simulate 50% optimal expansion of each network ◮ Regress optimal infrastructure growth on characteristics of each cell

Investment Population 0.104*** Income per Capita 0.007 Consumption per Capita 0.179*** Infrastructure

• 0.195***

Differentiated Producer 0.133*** R2 0.32

These observables explain ∼30% of spatial distribution of optimal investments

◮ Rest is geography and pre-existing links

Regional Effects (Spain)

Optimal Expansion

reallocation

Which regions grow?

Dependent variable: Employment growth in the counterfactual Reallocation Expansion Population

• 0.002
• 0.001

Income per Capita 0.001

• 0.002

Consumption per Capita

• 0.147***
• 0.139***

Infrastructure 0.002 0.005*** Infrastructure Growth 0.013* 0.032** Differentiated Producer 0.013** 0.023*** R2 0.57 0.67 These regressions pool the outcomes across all locations in the convex case.

Conclusion

We developed a framework to study optimal transport networks

1

Neoclassical model (with labor mobility) on a graph

2

Optimal Transport Flows with congestion

3

Optimal Network design Forces not (yet) included:

◮ Indirect effects through further investments (e.g., in building structures) ◮ Other factors (land, labor) in the infrastructure construction cost ◮ Investments in trade hubs ◮ Optimal network investments around second best (e.g., distortions) ◮ Agglomeration and spillovers in production ◮ Dynamics

Potential applications for future work

◮ Optimal urban network ◮ International trade facilitation ◮ Developing countries ◮ Political economy and competing planners ◮ Instruments for location of infrastructure

Planner’s Problem

Definition

The planner’s problem with labor mobility is W = max

Ijk

max

Cj ,Lj ,... max Qn jk

u subject to (i) availability of traded and non-traded goods, cjLj ≤ C T

j (Cj) and hjLj ≤ Hj for all j;

(ii) the balanced-flows constraint, C n

j +

• k∈N (j)
• Qn

jk + τ n jk

• Qn

jk, Ijk

• Qn

jk

• = F n

j

• Ln

j , ...

• +
• i∈N (j)

Qn

ij for all j, n;

(Multiplier: Pn

j )

(iii) free labor mobility, Lju ≤ LjU (cj, hj) for all j; (iv) the network building constraint,

• j
• k∈N (j)

δI

jkIjk = K

with bounds I jk ≤ Ijk ≤ I jk ; and (v) factor market clearing and non-negativity constraints.

back

Example: Role of Geography

Non-convex case

Building Cost: δI

jk = δ0Distanceδ1 jk

• 1 + |∆Elevation|jk

δ2 δ

CrossingRiverjk 3

δ

AlongRiverjk 4

back

Example: Spain

(a) Underlying Graph (b) Actual Road Network (c) Measured Infrastructure I obs

jk back

Regional Effects (Spain)

Optimal Reallocation

back